Introducing Basic Differential Operators

 Differential Operators are useful, and often shorthand ways to approach writing differential equations - including systems of equations. They can be written in various forms, but basically:

D _x u  =   ux   e.g. 

 

 du/ dx =   ux

Are first degree ordinary differential equations.  

But we begin by using basic forms, say to obtain derivatives.   In general:

(D _x y) ⁿ  =   dⁿ y / dxⁿ       n=  1, 2, 3….

Then:  (D _x y) ²  =   d² y/ dx²      

 

(D _x y) ³  =   d³ y/ dx³      

And so forth:

Examples:

a)  Find:  (D _x y) ²   (4x³  +  3x² ) 

Then:  d² y/ dx²   (4x³  +  3x² ) 

=  12 x²  +   6x

b)  (D _x y)  (4/ x)

Then:  d y/ dx   (4/x) =  -4/  x² 

c)  Find:  (D _x y) ²   (sin 4x)

Then:  d² y/ dx²   (sin 4x)  

Take 1st derivative first:

dy/ dx (sin 4x)  = 4 cos (4x)

Then take derivative again, i.e.

dy/ dx (4 cos (4x))

=  - 16 sin (4x)

Of special interest is the inverse differential operator, viz.

(D^-1  x ⁿ )   =    ( xⁿ⁺¹ )  /  (n + 1)

Similarly for higher order::

(D^-2  x ⁿ )   =  (D^-1 ) ( xⁿ⁺¹ )  /  (n + 1) = 

xⁿ⁺²  /  (n  +1 )  (n  +  2) 

And for higher order (k) in general:

(D^-k  x ⁿ )   =    (x^n+k )  /  (n+ 1) (n + 2)...(n  +   k)  

Thus,  1/D stands for the integral, e.g.   ∫  F(x)dx  but with denominator corrected for as given by inverse formulae above.

 

And is the inverse differential operator

 

And 1/ D ⁿ  stands for successive (n)  integration.

So we see the inverse operator has the same effect as integration, but disregarding the constants of integration.

Suggested Problems:

Evaluate each of the following:

1) (D _x y)  (ln e ^x / 1+  e ^x)

2) (D _x y)  ½ ( e ^x   -  e ^-x)   

3) D^-1  ( 11 x² )  

 4)  (D _x y)    ( e ^x   ln x)

5)  D^-1  (e ^{sin x}  cos x)  

6)  D^-1  (tanh x )  

7)  (D _x y)  (sech ² x)

8)   D^-2  (3 e ^x )  

9)  D^-1  ( cosh ²  3x)

10)   D^-2  (1/  x² ) 

11) D^-1  ( 3x² )

12) (D  - a) e ^ax

13) (D  - a)^-1   e ^ax

¹⁴⁾ (D- 1) ² x ³  e ^x 

15) (D  - 1)^-1   x⁴